Theorem wl-ifp-ncond1 35146

Description: If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.)

Assertion

Ref	Expression
wl-ifp-ncond1	⊢ (¬ 𝜓 → (if-(𝜑, 𝜓, 𝜒) ↔ (¬ 𝜑 ∧ 𝜒)))

Proof of Theorem wl-ifp-ncond1

Step	Hyp	Ref	Expression
1		df-ifp 1060	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
2		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
3	2	con3i 157	. . 3 ⊢ (¬ 𝜓 → ¬ (𝜑 ∧ 𝜓))
4		biorf 935	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓) → ((¬ 𝜑 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))))
5	3, 4	syl 17	. 2 ⊢ (¬ 𝜓 → ((¬ 𝜑 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))))
6	1, 5	bitr4id 294	1 ⊢ (¬ 𝜓 → (if-(𝜑, 𝜓, 𝜒) ↔ (¬ 𝜑 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845  if-wif 1059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-ifp 1060

This theorem is referenced by:  wl-ifp-ncond2  35147